IPhone 4s need to be backed up while on recovery mode?

I have an iphone 4s ios 7.02 and the phone only turns on when connected to the wall but when connected to a computer it goes to the apple loo then shuts off then to a charging logo. I can't backup the stuff when it turns on when plugged to the wall because there's a Wi-Fi problem where i can't connect to Wi-Fi and can't connect to a 3g network. So i don't have a backup of it. I need to back up the stuff without restoring or deleting it. Plus i have tried to restore but gives me error: 4031 or something like that. Its a hopeless case but i know there's a way i just want to connect it to the computer back it up and throw this phone away unless if there's another way but i don't want to replace the battery or anything

Reset the phone. While NOT connected to a computer hold down the power and home button at the same time until you get an Apple logo.

You may have to run recovery mode. That will wipe the phone. Basically do a reset while it is connected to a computer while iTunes is open.

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